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Given a compact Lie group $K$ and a maximal torus $T\leq K$, and choose a positive Weyl chamber $\mathfrak t^*_+\subset\mathfrak k^*$, where we used a $K$-invariant inner product on $\mathfrak k$. Then $\mathfrak t^*_+$ can be decomposed into disjoint open faces \begin{equation} \mathfrak t^*_+=\bigsqcup_{\sigma\in\Sigma}\sigma \end{equation} where $\Sigma$ is the set of all faces.

Then question is about centralizer $K_\xi$ of $\xi\in\sigma\in\Sigma$, do we have $K_\xi$ only depends on $\sigma$ and not depends on $\xi\in\sigma$?

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The identity component of the centralizer depends only on the face, but the component group is more sensitive. For proof of these statements, an example of the dependence, and an explanation of how to compute both, I refer to §2.4 of Mark Reeder's excellent survey, "Torsion automorphisms of simple Lie algebras".

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